The Stacks project

Lemma 22.36.6. Let $(A, \text{d})$ be a differential graded algebra. Assume that $A^ n = 0$ for $|n| \gg 0$. Let $E$ be an object of $D(A, \text{d})$. The following are equivalent

  1. $E$ is a compact object, and

  2. $E$ can be represented by a differential graded $A$-module $P$ which is finite projective as a graded $A$-module and satisfies $\mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M) = \mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(P, M)$ for every differential graded $A$-module $M$.

Proof. Let $\mathcal{D} \subset K(A, \text{d})$ be the triangulated subcategory discussed in Remark 22.36.1. Let $P$ be an object of $\mathcal{D}$ which is finite projective as a graded $A$-module. Then $P$ represents a compact object of $D(A, \text{d})$ by Remark 22.36.2.

To prove the converse, let $E$ be a compact object of $D(A, \text{d})$. Fix $a \leq b$ as in Lemma 22.36.5. After decreasing $a$ and increasing $b$ if necessary, we may also assume that $H^ i(E) = 0$ for $i \not\in [a, b]$ (this follows from Proposition 22.36.4 and our assumption on $A$). Moreover, fix an integer $c > 0$ such that $A^ n = 0$ if $|n| \geq c$.

By Proposition 22.36.4 we see that $E$ is a direct summand, in $D(A, \text{d})$, of a differential graded $A$-module $P$ which has a finite filtration $F_\bullet $ by differential graded submodules such that $F_ iP/F_{i - 1}P$ are finite direct sums of shifts of $A$. In particular, $P$ has property (P) and we have $\mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(P, M) = \mathop{\mathrm{Hom}}\nolimits _{K(A, \text{d})}(P, M)$ for any differential graded module $M$ by Lemma 22.22.3. In other words, $P$ is an object of the triangulated subcategory $\mathcal{D} \subset K(A, \text{d})$ discussed in Remark 22.36.1. Note that $P$ is finite free as a graded $A$-module.

Choose $n > 0$ such that $b + 4c - n < a$. Represent the projector onto $E$ by an endomorphism $\varphi : P \to P$ of differential graded $A$-modules. Consider the distinguished triangle

\[ P \xrightarrow {1 - \varphi } P \to C \to P[1] \]

in $K(A, \text{d})$ where $C$ is the cone of the first arrow. Then $C$ is an object of $\mathcal{D}$, we have $C \cong E \oplus E[1]$ in $D(A, \text{d})$, and $C$ is a finite graded free $A$-module. Next, consider a distinguished triangle

\[ C[1] \to C \to C' \to C[2] \]

in $K(A, \text{d})$ where $C'$ is the cone on a morphism $C[1] \to C$ representing the composition

\[ C[1] \cong E[1] \oplus E[2] \to E[1] \to E \oplus E[1] \cong C \]

in $D(A, \text{d})$. Then we see that $C'$ represents $E \oplus E[2]$. Continuing in this manner we see that we can find a differential graded $A$-module $P$ which is an object of $\mathcal{D}$, is a finite free as a graded $A$-module, and represents $E \oplus E[n]$.

Choose a basis $x_ i$, $i \in I$ of homogeneous elements for $P$ as an $A$-module. Let $d_ i = \deg (x_ i)$. Let $P_1$ be the $A$-submodule of $P$ generated by $x_ i$ and $\text{d}(x_ i)$ for $d_ i \leq a - c - 1$. Let $P_2$ be the $A$-submodule of $P$ generated by $x_ i$ and $\text{d}(x_ i)$ for $d_ i \geq b - n + c$. We observe

  1. $P_1$ and $P_2$ are differential graded submodules of $P$,

  2. $P_1^ t = 0$ for $t \geq a$,

  3. $P_1^ t = P^ t$ for $t \leq a - 2c$,

  4. $P_2^ t = 0$ for $t \leq b - n$,

  5. $P_2^ t = P^ t$ for $t \geq b - n + 2c$.

As $b - n + 2c \geq a - 2c$ by our choice of $n$ we obtain a short exact sequence of differential graded $A$-modules

\[ 0 \to P_1 \cap P_2 \to P_1 \oplus P_2 \xrightarrow {\pi } P \to 0 \]

Since $P$ is projective as a graded $A$-module this is an admissible short exact sequence (Lemma 22.16.1). Hence we obtain a boundary map $\delta : P \to (P_1 \cap P_2)[1]$ in $K(A, \text{d})$, see Lemma 22.7.2. Since $P = E \oplus E[n]$ and since $P_1 \cap P_2$ lives in degrees $(b - n, a)$ we find that $\mathop{\mathrm{Hom}}\nolimits _{D(A, \text{d})}(E \oplus E[n], (P_1 \cap P_2)[1])$ is zero. Therefore $\delta = 0$ as a morphism in $K(A, \text{d})$ as $P$ is an object of $\mathcal{D}$. By Derived Categories, Lemma 13.4.11 we can find a map $s : P \to P_1 \oplus P_2$ such that $\pi \circ s = \text{id}_ P + \text{d}h + h\text{d}$ for some $h : P \to P$ of degree $-1$. Since $P_1 \oplus P_2 \to P$ is surjective and since $P$ is projective as a graded $A$-module we can choose a homogeneous lift $\tilde h : P \to P_1 \oplus P_2$ of $h$. Then we change $s$ into $s + \text{d} \tilde h + \tilde h \text{d}$ to get $\pi \circ s = \text{id}_ P$. This means we obtain a direct sum decomposition $P = s^{-1}(P_1) \oplus s^{-1}(P_2)$. Since $s^{-1}(P_2)$ is equal to $P$ in degrees $\geq b - n + 2c$ we see that $s^{-1}(P_2) \to P \to E$ is a quasi-isomorphism, i.e., an isomorphism in $D(A, \text{d})$. This finishes the proof. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09RB. Beware of the difference between the letter 'O' and the digit '0'.